Suites 1. Déterminer toutes les suites réelles telles que : avec a = 3

2013 −2014
∀n∈N, un+2 + 5un+1 −14un=ana= 3 a= 2
u0=a∈Rn∈N
un+1 =−u2
n+ 5un−4
un+1 =(un−3)3
4un+1 =u2
n−2un+1 = cos un
n∈N∗Pn=
n
k=1
Xk−1
[0,1] an(an)
α an−α
x0=y0= 0 n
xn+1 =7−ynn+1 =√7−xn
L L′xn−L yn−L′
u0n un+1 =b+asin(un)
b∈Ra∈]−1,1[
(un)
un+1 −un(un)
(xn)n∈N∗
Sn=
n
k=1
xk(Sn) +∞
(un)Cvn
vn=n
k=1 xkuk
Sn
n∈N∗
(un)α(vn)
α
(un) +∞
(tn)tn∼xnTn=
n
k=1
tk
Tn∼Sn
u0
∀n∈Nun+1 =e−ununα
uα
n+1 −uα nun
un=
n
k=1
1
√k−2√n+ 1 vn=
n
k=1
1
√k−2√n

2013 −2014
(xn) lim
n→+∞xn+1
2x2n=ℓ ℓ
u0CnN
un+1 =1
2[un+|un|]
a, b
(an)n∈N(bn)n∈Nlim
n→+∞
an
n=alim
n→+∞
bn
n=b
p, q
(pan+qbn)n
u0, u1, u2n
un+3 =3
√unun+1un+2
(un) (u2n) (u2n+1) (u3n)
(un)
nN∗un=
n
k=0
1
k!vn=un+1 n×n!
e
n∈N∗λn]0,1[ e=un+λn
n×n!
e
a b (√2 + 1)n=an+√2bn
n∈NpnN
(√2 + 1)n=√pn+pn+ 1
(u)vp=sup{un;n > p}
(vn) (vn)
(un)−∞
(un) (vn)
(un) (vn) +∞(un)
un+1 =−4
un+ 4 un+1 =u2
n−un+ 1 un+1 =√1 + un
n an+2 ≤1
3(an+1 + 2an)
n
k=1
n3+k2
n3
n
k=1
n3−k2
n3
n
k=1
n
n2+k
n
k=1
n
n2+k21
0
tnsin tdt un=nsin(2πen!)